PTSP-UNIT II Q ue st io ns & A n sw e rs

GRIET‐ECE 1
UNIT-2
1. Explain in detail about cumulative distribution function (CDF)?
Ans:
CUMULATIVE DISTRIBUTION FUNCTION (CDF)
It is the probability that a random variable X takes values less than or equal to x and is
denoted as F
X
(x).

Thus, F
X
(x) = P ( X ≤ x) , where P ( X ≤ x) = ∑ P
i

x
i
≤ x
i.e., to get cumulative distribution function of a random variable X, add all the probabilities
with which 'X' takes all the values less than or equal to x.
Consider the random experiment of tossing a die. Let X be the random variable defined on
the sample space of the above experiment such that it takes values equal to the outcomes of the
experiments.
Thus, the probability distribution of X is
TABLE 1:

1. F
X
( x) ≥ 0 and 0 ≤ F
X
( x) ≤ 1 :
Since, CDF is sum of the probabilities and since probabilities are always non-negative (i.e.,
greater than or equal to zero), CDF is always non-negative.
It is numerically bounded between 0 and 1.

2.(i) F
X
( - ∞) = 0 :

As per the definition of CDF, F
X
(-∞) = P ( X ≤ - ∞) since X is having all the
assigned real values existing between - ∞ and + ∞, and since there are no real numbers
less than - ∞ , P( X ≤-∞ ) = 0, i.e., F
X
(-∞) = 0
(ii) F
X
( - ∞) = 1 :
As per the definition of CDF, F
X
(∞) = P ( X) i.e., it includes all the values taken by
X and the sum of the probabilities of X taking all the values less than or equal to ∞ is 1.

PROBABILITY DENSITY FUNCTION (PDF):
Probability density function (PDF) of a random variable X is defined as
f
X
(x) =
d
dx
F
X
(x).i.e., change in CDF is named as PDF.
Consider the CDF plot in fig 3.1
From x
i
= -∞ to x
i
=0, there is no change in CDF , and hence PDF=0.
At x
i
= 1, CDF is changed by 1/6. So, the Pdf of that random variable at x
i
= 1 is 1/6
From x
i
= 1 to x
i
= 2, there is no change in CDF and hence Pdf = 0.
At x
i
= 2, CDF is changed from 1/6 to 2/6 i.e., by 1/6, and so Pdf - 1/6.
Thus, the plot of Pdf is given as

Thus, the PDF of a discrete random variable is an impulse train. Thus ,the PDF of a discrete
random variable can be expressed

d
dx
F
X
(x) =
d
dx
|∑ P(x
ì
). u(x -x
ì
)
ì
]
∑ P(x
ì
). o(x -x
ì
)
ì
j
d
dx
u(x) = o(x)[
For a continuous random variable since CDF is a continuous function, its Pdf which is d/dx(CDF) is also a
continuous function, provided the derivative exists. In the CDF plot of a continuous random variable, at the points
where there is an abrupt change in the slope, the Pdf is plotted as a step-type discontinuity.

GRIET‐ECE 6
Consider the following probability distribution of a discrete of a discrete random variable X:
X=x
i

1 2 3
P(X
=x
i
)
1
/
3
1
/
3
1
/
3

Its PDF plot is

Here, Pdf at X = 1 is 1/3 and P ( X = 1) is also 1/3 . Similarly Pdf at X = 2 is same
P( X = 2) and so on.
Thus, for a discrete random variable, probability and probability density function are found to be the same.
So, to specify a discrete random variable, both of the above need not be specified.

F
X
(x)=]
x
6
4
2
dx =1
These limits give the value of F
X
( x) at ∞ i.e., at the largest value taken by X i.e., 4. But, the
CDF at remaining X between 2 and 4 can't be obtained.
(ii) So, to get an expression for CDF in terms of x, take the lower limits of the integration as
defined for X and upper limit as x

6. (a)Explain CDF and PDF for a random variable ?
(b)Explain condition distribution and state its properties
(c)What is conditional density .List its properties
ANS: (a) Cumulative Distribution Function (CDF):
This function is very close to relative frequency function which gives the probabilities of
different outcomes of a random experiment. The CDF is the probability less than or equal to
some specified value or outcome of a random experiment.
This is defined as
www.jntuworld.com
www.jntuworld.com
PTSP-UNIT II Q ue st io ns & A n sw e rs

GRIET‐ECE 11
where x
k
is some value that belongs to the random variable X. Note that, F
X
(x
k
) is a function of
x, not of the random variable X. For any point x
k
, the distribution function F
X
( x
k
) express a
probability.
Probability Density Function (PDF):
The probability density function is defined as
With reference to the following plot of the distribution function against the possible outcome
Therefore the probability of finding random variable x in infinitesimally small interval 'dx'
is given by
P( x ≤ X≤ ( x + dx)) = P
X
( x ) dx
Similarly probability of finding random variable between x
1
to x
2
is given by